Multiple-3D-object decryption based on one ... - OSA Publishing

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Research Article

Vol. 56, No. 22 / August 1 2017 / Applied Optics

Multiple-3D-object decryption based on one interference using two phase-only functions WEI-NA LI,1 SANG-MIN LEE,1 SANG KEUN GIL,2

AND

NAM KIM1,*

1

Department of Information & Communication Engineering, Chungbuk National University, 1 Chungdae-ro, Seowon-gu, Cheongju-si, Cheongju 362-763, South Korea 2 Department of Electronic Engineering, The University of Suwon, Whasung 440-600, South Korea *Corresponding author: [email protected] Received 12 April 2017; revised 22 June 2017; accepted 22 June 2017; posted 22 June 2017 (Doc. ID 292753); published 26 July 2017

We propose a multiple-3D-object decryption scheme based on one interference using two beams that are from two different phase-only functions. It takes advantage of off-axis digital holography to extract the optical fields of multiple 3D objects, and respectively utilize single different decryption keys to decrypt multiple 3D objects in the decryption procedure. The advantages of the proposed scheme include the following: each 3D object can be decrypted discretionarily without decrypting a series of other objects earlier; no iterative algorithm is involved; and the decrypted image of each object can be successfully clearly distinguished. The feasibility of the proposed scheme is verified by the optical holograms of real 3D objects. © 2017 Optical Society of America OCIS codes: (090.0090) Holography; (090.1995) Digital holography; (090.2880) Holographic interferometry; (100.4998) Pattern recognition, optical security and encryption; (260.3160) Interference. https://doi.org/10.1364/AO.56.006214

1. INTRODUCTION Refregier and Javidi first suggested the double random phase encoding (DRPE) method in the 1990s [1], which accelerated the development of optical information encryption techniques in recent decades. On the other hand, Vu et al. first suggested a kinogram-based single-phase decryption technique [2], where only one spatial light modulator (SLM) is used to simultaneously display the encrypted information and the decryption key. This technique requires a complicated experimental setup and elaborate alignment. They also improved the optical decryption system based on kinogram encoding afterwards [3], which is simpler and more robust. Li et al. proposed a method to encrypt one image utilizing two phase-only functions (POFs) to completely remove the silhouette problem [4]. Subsequently, they expanded a multiple-image encryption (MIE) method based on five POFs [5]. Gil and Jeon lately suggested two novel asymmetric cryptosystems based on free-space interconnected optical logic operations to enhance security level [6,7]. They also proposed a new optical one-time password authentication using two-step phase-shifting digital holography to guard against a reply attack and enhance security level [8]. However, optical image encryption techniques are monotonous on the display of two-dimensional (2D) images; optical encryption of three-dimensional (3D) scenes is gradually becoming a subject to be investigated. Tajahuerce and Javidi proposed a method for optical encryption of 3D information 1559-128X/17/226214-08 Journal © 2017 Optical Society of America

utilizing the four-step phase-shifting method [9]. Matoba and Javidi proposed an optical 3D display system interfaced with digital data transmission [10]. Cho and Javidi proposed a 3D photon counting technique using passive integral imaging [11]. Lee and Cho proposed a method for optical encryption and information authentication of a 3D object to effectively mitigate wireless channel effects [12]. Muniraj et al. suggested a method for 3D scene acquisition via reconstruction with multispectral information and Fourier-based encryption using computational integral imaging [13]. Nevertheless, the five papers are all for encrypting only a single 3D object. In this paper, we propose a multiple-3D-object decryption scheme to decrypt multiple 3D objects from one mutual ciphertext by utilizing single different decryption keys. This scheme is based on one interference of two beams that are from two different phase-only functions. It takes advantage of offaxis digital holography to extract the optical fields of multiple 3D objects, and respectively utilize single different decryption keys to decrypt multiple 3D objects in the decryption procedure. Compared with our previous paper [5], the intention is to decrease five POFs to two POFs; moreover, we attempt to extend MIE to a multiple-3D-object cryptosystem by utilizing off-axis digital holography. To the best of our knowledge, it is a novel decryption scheme to decrypt multiple 3D objects based on one interference using two POFs. In addition, the advantages of the proposed scheme also remain: each 3D object can be decrypted discretionarily without decrypting a series of

Research Article

Vol. 56, No. 22 / August 1 2017 / Applied Optics

other objects earlier, and no iterative algorithm is involved. Eventually, one of the multiple 3D objects can be successfully decrypted by one mutual ciphertext and one corresponding decryption key (both of them are POFs), which is the most important enhancement. The rest of this paper is organized as follows. Preliminary theoretical principles are introduced in Section 2. In Section 3, the feasibility demonstrations of the preliminary decryption scheme are presented by the experimental results. In Section 4, the feasibility of the proposed decryption scheme is verified. Finally, the conclusions are summarized in Section 5.

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Fig. 1. Flowchart of production of the normalized complex wave field distribution CmCp
x; y.

2. PRELIMINARY THEORETICAL PRINCIPLES A. Encryption Procedure

Assume one 3D object OBJ 1
ξ; η is Fresnel transformed to the spatial frequency domain with an illumination wavelength λ and propagation distance z 1 , which are represented by F RT λ fOBJ 1
ξ; η; z 1 g. A complex optical field H 1
x; y can be obtained: H 1
x; y  F RT λ fOBJ 1
ξ; η; z 1 g;

(1)

where F RT λ f·g represents the Fresnel transform with an illumination wavelength λ introduced in detail by Goodman [14]. Throughout the remainder of this paper, the coordinates
ξ; η and
x; y represent the object domain and the spatial frequency domain, respectively. By applying the same process to the remaining n − 1 3D objects OBJ 2
ξ; η; OBJ 3
ξ; η; …, and OBJ n
ξ; η, the optical fields H 2
x; y; H 3
x; y; …, and H n
x; y are obtained. Note that each 3D object is Fresnel transformed with a different distance. Summation of the entire set of the optical field distribution is given by CmCp_tmp1
x;yH 1
x;yH 2
x;yH n
x;y: (2) In order to reduce the correlation between each POF and the corresponding complex amplitude H n
x; y, a fixed random complex amplitude randx, in which the maximum of Ax is the maximum amplitude of CmCp_tmp1
x; y, is added to CmCp_tmp1
x; y to acquire CmCp_tmp2
x; y, which are shown in Eqs. (3) and (4). Meanwhile, the cross-talk noise CrT 1_tmp
x; y for decrypting OBJ 1
ξ; η, the summation of the optical fields H 2
x; y; H 3
x; y; …; H n
x; y, and randx is expressed in Eq. (6). Since the domain of cos −1
· is [−1, 1], the maximum value max_cmcp of abs
CmCp_tmp2
x; y (absf·g represents the complex modulus) is introduced to Eqs. (5) and (7) to normalize the amplitude of CmCp_tmp2
x; y and CrT 1_tmp
x; y, respectively. The cross-talk noises of the other objects should also be normalized surely. The flowchart for the production of CmCp
x; y is depicted in Fig. 1. rand x  Ax exp
i  θx ; CmCp_tmp2
x; y  CmCp_tmp1
x; y  rand x; CmCp
x; y 

CmCp_tmp2
x; y ; max _cmcp

(3) (4) (5)

CrT 1_tmp
x; y  H 2
x; y      H n
x;y  rand x; CrT 1
x; y 

CrT 1_tmp
x; y : max _cmcp

(6)

(7)

However, the image sensor cannot fully capture the optical field (complex amplitude) of the 3D scene in the optical experiment. Therefore, the digital holography technique, which involves recording the complex amplitude distribution of a 3D object using an image sensor and reconstructing an object image using a computer [15], is utilized to obtain the complex amplitude of the 3D scene. In the proposed scheme, the off-axis digital holographic interferometry is adopted to obtain the optical field of each 3D object, which implies that summation CmCp
x; y and the corresponding cross-talk noise of each object can both be obtained by the off-axis digital holographic method, for instance, CrT 1
x; y. Assume arn
x;y; θn  denotes the reference beam of OBJ n
ξ; η with a tilted angle θn . Hologram I n
x; y of OBJ n
ξ; η can be captured, which is mathematically expressed in Eq. (8). jarn
x; y; θn j2  jH n
x; yj2 denotes the zero-order image, H n
x; y × arn
x; y; θn  denotes the virtual image, and H n
x; y × arn
x; y; θn  denotes the real image, where  denotes a complex conjugate. The real image and virtual image are called twin images. Since the real image and the virtual image are separated in the frequency domain of I n
x; y by θn, the real/virtual image portion is cropped and zero-padded the same size as the original hologram. Subsequently, it is inverse Fourier transformed to obtain H n
x; y. I n
x; y  jarn
x; y; θn j2  jH n
x; yj2  H n
x; y × arn
x; y; θn   H n
x; y × arn
x; y; θn :

(8)

Equations (9)–(12) are employed to produce pof 1
x; y, which is the merely one ultimate mutual ciphertext for decrypting all of the 3D objects, and pof 2
x; y, which is to be eliminated at the end of the encryption processing. The sum of pof 1
x; y and pof 2
x; y equals CmCp
x; y, which can be proven by Euler’s formula. It is shown in Eq. (13): ACp
x; y  absfCmCp
x; yg;

(9)

PCp
x; y  argfCmCp
x; yg;

(10)

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  ACp
x;y ; (11) pof 1
x;yexp i  PCp
x;y−cos 2 



−1

    ACp
x;y ; (12) pof 2
x;yexp i  PCp
x;ycos−1 2 CmCp
x; y  pof 1
x; y  pof 2
x; y;

(13)

where absf·g represents the complex modulus, and argf·g represents the phase angle. The complex opposite of CrT 1
x; y is added to pof 2
x; y to derive the cross-talk eliminator CrT 1_el im
x; y, shown in Eq. (14). pof 3
x; y and pof 4
x; y are derived by employing Eqs. (15)–(18), which are two decryption keys in the preliminary decryption scheme. The sum of pof 3
x; y and pof 4
x; y equals CrT 1_tmp
x; y, which can be proven by Euler’s formula. It is shown in Eq. (19). Subsequently, pof 3
x; y and pof 4
x; y are directly transmitted to the corresponding authorized user to decrypt OBJ 1
ξ; η, and the transmission security will be demonstrated in Section 3. Meanwhile, the cross-talk noise CrT1(x,y) and cross-talk eliminator CrT 1_el im
x; y are both destroyed. The most serious issue in the proposed scheme is cross-talk noise, which is the same as MIE. Cross-talk noise renders the naked eye unable to distinguish the corresponding decrypted 3D object. The drawback of the cross-talk noise can be employed by using its complex opposite as the decryption keys. CrT 1_el im
x; y  −CrT 1
x; y  pof 2
x; y;

(14)

ACr
x; y  absfCrT 1_el im
x; yg;

(15)

PCr
x; y  argfCrT 1_el im
x; yg;

(16)

    ACr
x;y ; (17) pof 3
x;yexp i  PCr
x;y−cos−1 2     ACr
x;y −1 ; (18) pof 4
x;yexp i  PCr
x;ycos 2 CrT 1_el im
x; y  pof 3
x; y  pof 4
x; y:

(19)

be obtained using the preliminary decryption scheme. The mathematical expression for obtaining Re_obj1 (the decrypted OBJ 1
ξ; η), which also uses distance z 1 as a key, is given by Eq. (20) where IF RT λ f·g stands for the inverse Fresnel transform with an illumination wavelength λ. Re_obj1  abs
I F RT λ f
pof 1  pof 3  pof 4; z 1 g: (20) In the encryption process, the complex amplitude of each object in the spatial frequency domain is obtained by an off-axis digital holographic experiment. In the experimental results of the decryption process, the correlation coefficient (CC) is introduced as a criterion for illustrating the similarity between the decrypted image and the reconstructed image of the complex amplitude of each object in the spatial frequency domain. The correlation coefficient is defined by ρ

EfO − EO × R − ER g ; fEfO − EO 2 g × EfR − ER 2 gg2

(21)

where Ef·g denotes the expected value operator; O denotes the reconstructed image of the complex amplitude of each object in the spatial frequency domain, e.g., the reconstructed image of H 1
x; y; and R denotes the decrypted image of each object, e.g., Re_obj1 and Re_obj1 (it will be introduced later). Usually when ρ ≤ 0.2, it indicates the correlation is very weak; theoretically, people cannot distinguish the decrypted image with the naked eye. 3. FEASIBILITY DEMONSTRATIONS OF PRELIMINARY DECRYPTION SCHEME A. Experimental Results of the Preliminary Decryption Scheme

In this section, the feasibility of the preliminary decryption scheme is demonstrated by the optical experimental results. 12 non-transparent 3D objects with a height between 0.6 cm and 1.1 cm and a width between 0.5 cm and 1 cm, most of which are presented in Fig. 2(a). Figure 2(b) is a white toy mouse head with the size 0.5 cm by 0.6 cm, which denotes Object 1. The hologram was captured by a complementary metal oxide semiconductor (CMOS) image sensor with a resolution of 2048 × 2048 pixels, and a pixel pitch of 5.5 μm. The optical experimental setup is shown in Fig. 3. The hologram of Object 1 is captured first, as shown in Fig. 4(a), and the Fourier transform (FT) of the hologram in the frequency domain is presented in Fig. 4(b). Three lighter portions in the center of Fig. 4(b) can be observed,

Obviously, how much information of H 1
x; y that each of pof 1
x; y, pof 3
x; y and pof 4
x; y respectively contains is unknown; however, the specific verified data will be presented later. B. Preliminary Decryption Procedure

At the end of the encryption procedure, one ciphertext, pof 1
x; y, and two decryption keys, pof 3
x; y and pof 4
x; y, which are all POFs, are yielded. Since the summation of these POFs contains as much information as the H 1
x; y of the original 3D object OBJ 1
ξ; η in the spatial frequency domain, a high-quality decrypted OBJ 1
ξ; η can

Fig. 2. 3D objects, (a) most of the objects utilized in this experiment, and (b) Object 1, a white toy mouse head.

Research Article

Fig. 3. Optical experimental setup of the off-axis recording configuration: OBJ, real 3D object; BS, non-polarizing beam splitter; PBS, polarizing beam splitter; HWP, half-wave plate; M, mirror, CMOS, complementary metal oxide semiconductor image sensor.

where the portions surrounded by a red dashed box, a magenta dashed box, and green dashed box present the real image, the zero-order image, and the virtual image, respectively. Subsequently, the real image, which is located from 752 to 1296 vertically and from 300 to 844 horizontally, is cropped and zero-padded the same size as the original hologram, which is presented in Fig. 4(c). Then it is inverse Fourier transformed to obtain the complex amplitude H1, which is required in the preliminary scheme. Finally, the reconstructed image is obtained by the inverse Fresnel transform (IFT) of H1, shown in Fig. 4(d). All of the holograms of the 12 objects, which are shown in Fig. 5, were captured from different distances presented in Table 2. The complex amplitude of each object in the spatial frequency domain can be calculated by applying the same process to each hologram as the manner of the flow chart in Fig. 4. Since the complex amplitude of each object’s real image is obtained from the above process, the summation CmCp can be obtained by adding up the 12 complex amplitudes and one fixed random complex amplitude RANDX, where the normalization is included. Note that once RANDX is assigned, it will be identified to decrypt each object. Then CmCp is divided into two noise-like POFs (POF1 and POF2). POF1 is the

Fig. 4. Flow chart to obtain the reconstructed image from the hologram of Object 1 using the off-axis optical experimental setup: (a) hologram of Object 1, (b) Fourier transform of (a), (c) zero-padded image of the real image, and (d) reconstructed image of Object 1.

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ultimate mutual ciphertext for decrypting all 12 objects. On the other hand, the intermediate phase-only function POF2 is a portion of the decryption keys, which is to be eliminated at the end of the encryption procedure. When decrypting Object 1, CrT1 (the cross-talk noise of Object 1) is the summation of the same RANDX and the complex amplitudes of all the objects except for Object 1, in which the normalization is included as well. Then, the complex opposite of CrT1 is added to POF2 to obtain the cross-talk eliminator CrT1_elim for decrypting Object 1. Then CrT1_elim is divided into two decryption keys, POF3_obj1 and POF4_obj1. Subsequently, POF3_obj1 and POF4_obj1 are directly transmitted to the corresponding authorized user. Meanwhile, the cross-talk noise CrT1 and cross-talk eliminator CrT1_elim are both destroyed. In the decryption process, the mutual ciphertext POF1 is combined with decryption keys POF3_obj1 and POF4_obj1, and correct distance 60.5 cm, then the correct decrypted image can be obtained, which is presented in Fig. 6. It takes 12 s to obtain the mutual ciphertext and the corresponding decryption keys using the preliminary decryption scheme. The computer configuration includes a Windows 7 64-bit operating system, MATLAB R2012b, and an Intel Core i5-4460 CPU @3.20 GHz. In addition, the decrypted images of each object, which are generated from correct POF1, POF3, POF4, and decryption distance, are all presented in Fig. 7, where we can observe that each image is clearly distinguished. The CC values of the decrypted images in Fig. 7 are presented in Table 1. B. Robustness Analysis

At first, we show the reconstructed images generated merely from one phase-only function. Figure 8(a) presents the reconstructed image generated from only POF1. Figure 8(b) is the reconstructed image generated from only POF3_obj1. And Fig. 8(c) presents the reconstructed image generated from only POF4_obj1. The three images were all reconstructed with correct distances. It can be observed that Figs. 8(a)–8(c) are all noise-like images that cannot be distinguished. In Table 1, it is observed that the CC values of Figs. 8(a)–8(c) are 0.012, 0.013, and 0.011, respectively, which are all much less than 0.2. Subsequently, in Fig. 9, we show the decrypted images without mutual ciphertext POF1 or with fewer decryption keys. Figure 9(a) shows the decrypted image generated from only POF1 and POF4_obj1. Figure 9(b) presents the decrypted image generated from only POF3_obj1 and POF4_obj1. Figure 9(c) shows the decrypted image from only POF1 and POF3_obj1. The decryption distances were all correct. It can be observed that the decrypted images in Figs. 9(a)– and 9(c) are all noise-like images that cannot be distinguished. Moreover, their CC values are 0.01, 0.012, and 0.014, respectively, which are all much less than 0.2 as well. Moreover, the CC values of each object’s decrypted images generated from different combinations of the POFs were evaluated, which are all much less than 0.2, except the CC values of the decrypted images generated from POF1, POF3, and POF4, which are all not less than 0.99. These are all presented in Table 1. It indicates that the information of each object is

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Table 1.

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Vol. 56, No. 22 / August 1 2017 / Applied Optics

CC Values of Each Object Generated from Different POFs (POF1, POF3, and POF4)

Objects

POF1

POF3

POF4

POF1
POF4

POF3
POF4

POF1
POF3

POF1
POF3
POF4

Object Object Object Object Object Object Object Object Object Object Object Object

0.012 0.017 0.01 0.011 0.019 0.012 0.021 0.012 0.01 0.011 0.013 0.012

0.013 0.018 0.01 0.012 0.021 0.013 0.023 0.013 0.01 0.011 0.015 0.012

0.011 0.015 0.01 0.011 0.016 0.012 0.018 0.012 0.009 0.01 0.012 0.011

0.01 0.014 0.01 0.01 0.014 0.011 0.016 0.011 0.009 0.01 0.01 0.01

0.012 0.017 0.01 0.011 0.019 0.012 0.021 0.012 0.01 0.011 0.013 0.011

0.014 0.02 0.011 0.012 0.024 0.014 0.027 0.013 0.011 0.012 0.017 0.014

1.0 1.0 0.99 0.99 1.0 1.0 1.0 1.0 0.99 0.99 0.99 1.0

1 2 3 4 5 6 7 8 9 10 11 12

The CC values of the reconstructed images that are generated from POF3, POF4, and the summation of POF3 and POF4 are all much less than 0.2 in terms of Table 1. It indicates these reconstructed images are all white-noise images that cannot be distinguished. Therefore, it is very safe to directly transmit the decryption keys POF3 and POF4 to the corresponding authorized users. 4. FEASIBILITY VERIFICATIONS OF THE PROPOSED DECRYPTION SCHEME A. Proposed Decryption Scheme

Fig. 5. Captured holograms of each object using off-axis digital holography.

The alignment issue of three POFs’ interference is more severe in the optical experiment, compared with two POFs’ interference. Moreover, more components, e.g., SLM, non-polarizing beam splitter, mirrors, etc., are required. In order to decrease the storage space and the complexity of the optical decryption setup, there is an alternative to utilize two POFs instead of three POFs to decrypt one object. Since the reconstructed images that are generated from the sum of POF3 and POF4 are all noise-like images with CC values much less than 0.2 according to Table 1, the phase portion of the summation of pof 3
x; y and pof 4
x; y is extracted as a decryption key (POF), which is derived from Eqs. (22) and (23). The summation also equals the cross-talk eliminator CrT1_elim. Obviously, how much information of H 1
x; y that dkey_obj1 contains is unknown, as well. d keyp_obj1  argfpof 3  pof 4g  argfCrT 1_el im
x; yg; d key_obj1  exp
i  d keyp_obj1 :

Fig. 6. Decrypted image of Object 1 generated from correct mutual ciphertext POF1, decryption keys POF3_obj1 and POF4_obj1, and the correct decryption distance.

not equally divided into three portions as POF1, POF3, and POF4 in terms of Table 1. In addition, the reconstructed images that are generated from the sum of POF3 and POF4 are all noise-like images with CC values much less than 0.2.

(22) (23)

The decrypted image generated from POF1 and DKEY_obj1 with the correct distance can be clearly distinguished with the naked eye, which is shown in Fig. 10. Moreover, its CC value equals 0.529, which means a moderate correlation when 0.4 < CC ≤ 0.6. Therefore, the decrypted images of each object, generated from correct POF1 and DKEY, are all presented in Fig. 11, where we can observe that the quality of each image is good enough to be clearly distinguished. Compared with the decrypted images of each object generated from correct POF1, POF3, and POF4, shown in Fig. 7, the quality of each

Research Article

Fig. 7. Correct decrypted images of each object using the preliminary decryption scheme.

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Fig. 10. Decrypted image of Object 1 generated from POF1 and DKEY_obj1 with the correct distance.

Table 2. CC Values of Each Object Calculated from Mutual Ciphertext (POF1) and Decryption Key (DKEY) Using the Proposed Decryption Scheme

Object Fig. 8. Reconstructed images: (a) generated from only POF1, (b) generated from only POF3_obj1, and (c) generated from only POF4_obj1.

Fig. 9. Decrypted images: (a) generated from only POF1 and POF4_obj1, (b) generated from only POF3_obj1 and POF4_obj1, and (c) generated from only POF1 and POF3_obj1.

Object Object Object Object Object Object Object Object Object Object Object Object

1 2 3 4 5 6 7 8 9 10 11 12

POF1

DKEY

POF1
DKEY

Reconstruction Distance (unit: cm)

0.012 0.017 0.01 0.011 0.019 0.012 0.021 0.012 0.01 0.011 0.013 0.012

0.012 0.017 0.01 0.011 0.018 0.012 0.021 0.012 0.01 0.011 0.013 0.011

0.529 0.511 0.487 0.462 0.55 0.484 0.543 0.485 0.48 0.526 0.521 0.495

60.5 44.1 40.2 47.9 48.1 53.5 53 57.3 35 63.1 65.4 67.9

one decryption key DKEY, which are both POFs. The schematic diagram of the proposed decryption setup for decrypting Object 1 is presented in Fig. 12, in which the ciphertext and the decryption key are displayed on two same spatial light modulators, respectively. Re_obj1  abs
IF RT λ f
pof 1  d key_obj1 ; z 1 g:

corresponding image in Fig. 11 is moderate. The CC value of each decrypted image is more than 0.46, and the mean CC value of the 12 objects is around 0.51, according to Table 2. It is noted that each object can be successfully decrypted by one mutual ciphertext POF1 and one corresponding decryption key DKEY. Therefore, three POFs are successfully decreased to two POFs in the proposed decryption scheme. The proposed decryption scheme is mathematically expressed in Eq. (24), where Re_obj1 denotes an approximation to Re_obj1 . It indicates that the information of each object can be successfully decrypted by utilizing one ciphertext POF1 and

(24)

In the proposed decryption scheme, DKEY is transmitted only to the corresponding authorized user to decrypt each object. Subsequently, the corresponding cross-talk noise and cross-talk eliminator are both destroyed. It is not necessary to generate POF3 and POF4 as well. B. Robustness Analysis and Discussion

The reconstructed image generated from only POF1 is shown in Fig. 8(a). Figure 13(a) presents the reconstructed image generated from only DKEY_obj1. Subsequently, Fig. 13(b) shows the reconstructed image generated from correct POF1 and

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Table 3. CC Values of the Decrypted Image of Object 1 Generated from Incorrect POF1 or DKEY_obj1 CC of the Decrypted Image Only DKEY_obj1 is incorrect Only POF1 is incorrect

0.007 0.006

Fig. 11. Correct decrypted images of each object using the proposed decryption scheme.

Fig. 14.

Fig. 12. Schematic diagram of the proposed decryption setup. HWP, half-wave plate; L, lens; M, mirror; SF, spatial filter; BS, non-polarizing beam splitter; SLM, spatial light modulator; CCD, charge-coupled device camera.

incorrect DKEY_obj1, whose phase is random. On the other hand, Fig. 13(c) presents the decrypted image generated from correct DKEY_obj1 and incorrect POF1, whose phase is random. The four images were all generated with correct distances. It can be observed that Figs. 8(a) and 13(a)–13(c) are all noiselike images that cannot be distinguished. In terms of Tables 2 and 3, the CC values of Figs. 8(a) and 13(a)–13(c) are 0.012, 0.012, 0.007, and 0.006, respectively, which are all much less than 0.2. For the sake of illustrating the relation between the CC value and the decryption distance, Fig. 14 presents the

Distance-CC curve for the decrypted images of Object 1.

sensitivity of the CC value along with variations in decryption distance for the decrypted image of Object 1 when POF1 and DKEY_obj1 are both correct. As introduced previously, the correct distance is 60.5 cm. At distances from 1 cm to 40.2 cm, the CC value of the decrypted image is under 0.1, shown as an oscillating curve. From 40.2 cm to 60.5 cm, the CC curve is shown as a steep curve from 7 × 10−5 , which reaches 0.529 when the decryption distance is equal to the correct distance (60.5 cm). However, the curve decreases quickly from 60.5 cm to 80.8 cm. The CC value reaches 8 × 10−5 at 80.8 cm. The CC curve similarly shows an oscillation under 0.1 after 80.8 cm. It can be observed that the farther the decryption distance is from the correct distance, the lower the quality of the decrypted image of the 3D object. Even though the decryption distance cannot be considered an independent key, the obstruction capability is drastically enhanced as the decryption distance becomes farther from the correct distance. Therefore, an accurate decryption distance is helpful for the robustness of the proposed scheme. 5. CONCLUSION

Fig. 13. The image generated from only DKEY_obj1, the decrypted images: (b) generated from correct POF1 and incorrect DKEY_obj1, and (c) generated from incorrect POF1 and correct DKEY_obj1.

In this paper, we propose a multiple-3D-object decryption scheme to decrypt multiple 3D objects from one mutual ciphertext by utilizing single, different decryption keys. The proposed scheme is based on one interference of two beams that are from different phase-only functions. It takes advantage of off-axis digital holography to extract the optical fields of

Research Article multiple 3D objects, and respectively utilize single different decryption keys to decrypt multiple 3D objects in the decryption procedure. Compared with our previous method, one of the two ultimate ciphertexts is considered a portion of the decryption keys, which helps to get rid of POF2. Nevertheless, the decrypted image generated from the mutual ciphertext POF1 and the phase portion of the corresponding cross-talk eliminator can also be clearly distinguished with the naked eye. The CC values show that the correlation is moderate between the decrypted image and the correctly reconstructed image. Therefore, five POFs are decreased to two POFs. Namely, each of the multiple 3D objects can be successfully decrypted by utilizing one mutual ciphertext (POF1) and one corresponding decryption key (DKEY), which is the most important enhancement. To the best of our knowledge, it is a novel decryption scheme to decrypt multiple 3D objects based on one interference using two POFs. In addition, the advantages of the proposed scheme also ensure that each 3D object can be decrypted discretionarily without decrypting a series of other objects earlier; and no iterative algorithm is involved. Moreover, since the off-axis digital holographic method is introduced, the proposed scheme can be extended to multiple-3D-object cryptosystems. Funding. Ministry of Science, ICT and Future Planning (MSIP), Korea, under the ITRC (Information Technology Research Center) support program (IITP-2017-2015-0-00448). REFERENCES 1. P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995).

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